Let $G$ be a connected, simply connected, compact semisimple Lie group of dimension $n$. It has been shown by Clerc \cite{ Clerc1974} that, for any $f\in L^1(G)$, the Bochner-Riesz mean $S_R^\delta(f)$ converges almost everywhere to $f$, provided $\delta>(n-1)/2$. In this paper, we show that, at the critical index $\delta=(n-1)/2$, there exists an $f\in L^1(G)$ such that $$\limsup_{ R\rightarrow\infty} \big|S_{ R}^{ (n-1)/2}(f)(x)\big|=\infty, \ \text{ a.e.}\ x\in G.$$ This is an analogue of a well-known result of Kolmogorov \cite{ Kolmogoroff1923} for Fourier series on the circle, and a result of Stein \cite{ Stein1961} for Bochner-Riesz means on the tori $\mathbb T^{ n}, n\geq 2$. We also study localization properties of the Bochner-Riesz mean $S_{ R}^{ (n-1)/2}(f)$ for $f\in L^1(G)$.
Categories: math.CA